Actual problems of the approximation theory in metrics of discrete spaces on sets of summable periodic and almost periodic functions

This review paper highlights the main aspects of the development of research related to the solution of extreme problems in the theory of approximation in the spaces ${\mathcal{S}}^p$ and $B{\mathcal{S}}^p$ of periodic and almost periodic summable functions, respectively, where the $l_p$-norms of the sequences of Fourier coefficients are finite. In particular, the review contains the results known so far concerning the best, best $n$-term approximations and widths of classes of functions of one and many variables defined by means of $\psi$-derivatives and generalized moduli of smoothness in the spaces ${\mathcal{S}}^p$ and $B{\mathcal{S}}^p$. Particular attention is paid to the development of studies related to the derivation of direct and inverse approximation theorems in these spaces.